# Several properties related to interior and boundary of a set

We have the following propositions, which at the first glance seem to be true, but the truth still needs some careful consideration: If $U$ is open, $U=\Int(\overline{U})$. $\Bd(\Bd(A)) = Bd(A)$. $\Bd(\Int(A)) = \Bd(\overline{\Int(A)})$. For the first proposition, $\Int(\overline{U})$ represents the union of all the open sets which are contained in $\overline{U}$. $\because U \subset \overline{U}$ and $U$ is open, $\therefore U \subset \Int(\overline{U})$. If the inverse of this proposition is true, it implies that $U$ is the maximum among the open sets that are contained…